Falconer's conjecture

<p>

In <a href="/facts/Geometric_measure_theory/NLIQxYf9">geometric measure theory</a>, Falconer's conjecture, named after <a href="/facts/Kenneth_Falconer_(mathematician)/zf1e1BIG">Kenneth Falconer</a>, is an <a href="/facts/Open_problem/VuR6yLsW">unsolved problem</a> concerning the sets of <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distances</a> between points in <a href="/facts/Compact_space/d0cgXJH7">compact</a> 
  
    
      
        d
      
    
    {\displaystyle d}
  
-dimensional spaces. Intuitively, it states that a set of points that is large in its <a href="/facts/Hausdorff_dimension/rtX9ryrp">Hausdorff dimension</a> must determine a <a href="/facts/Distance_set/PHUuLIze">set of distances</a> that is large in <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a>. More precisely, if 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a <a href="/facts/Compact_set/d0cgXJH7">compact set</a> of points in 
  
    
      
        d
      
    
    {\displaystyle d}
  
-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> whose Hausdorff dimension is strictly greater than 
  
    
      
        d
        
          /
        
        2
      
    
    {\displaystyle d/2}
  
, then the <a href="/facts/Conjecture/oPWUb7SF">conjecture</a> states that the set of distances between pairs of points in 
  
    
      
        S
      
    
    {\displaystyle S}
  
 must have nonzero <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a>.
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Falconer's conjecture open-in-new

Falconer's conjecture